Syllogism

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.[3]

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions (axioms), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.

In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature.[9] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation and leads to discovering and building on axioms to create a more general conclusion.[9] Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics.

In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Although there were alternative systems of logic elsewhere, such as Avicennian logic or Indian logic, Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.

A noteworthy exception is the logic developed in Bernard Bolzano's work Wissenschaftslehre (Theory of Science, 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, because of the intellectual environment at the time in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.

This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.

One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the Roman Rota, which still requires that any arguments crafted by Advocates be presented in syllogistic format.

A categorical syllogism consists of three parts:

1. Major premise
2. Minor premise
3. Conclusion

Each part is a categorical proposition, and each categorical proposition contains two categorical terms.[13] In Aristotle, each of the premises is in the form "All A are B," "Some A are B", "No A are B" or "Some A are not B", where "A" is one term and "B" is another:

• "All A are B," and "No A are B" are termed universal propositions;
• "Some A are B" and "Some A are not B" are termed particular propositions.

More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, this is the minor term (i.e., the subject of the conclusion). For example:

Major premise: All humans are mortal.

Minor premise: All Greeks are humans.

Conclusion: All Greeks are mortal.

Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term; in this example, humans. Both of the premises are universal, as is the conclusion.

Major premise: All mortals die.

Minor premise: All men are mortals.

Conclusion: All men die.

Here, the major term is die, the minor term is men, and the middle term is mortals. Again, both premises are universal, hence so is the conclusion.

Relationships between the four types of propositions in the square of opposition

(Black areas are empty,
red areas are nonempty.)

There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M – Middle, S – subject, P – predicate.):

Major premise: All M are P.

Minor premise: All S are M.

Conclusion: All S are P.

The premises and conclusion of a syllogism can be any of four types, which are labeled by letters[14] as follows. The meaning of the letters is given by the table:

In Prior Analytics, Aristotle uses mostly the letters A, B, and C (Greek letters alpha, beta, and gamma) as term place holders, rather than giving concrete examples. It is traditional to use is rather than are as the copula, hence All A is B rather than All As are Bs. It is traditional and convenient practice to use a, e, i, o as infix operators so the categorical statements can be written succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic:

Form	Shorthand	Predicate logic
All A is B	AaB	${\displaystyle \forall x(A(x)\rightarrow B(x))}$  or  ${\displaystyle \neg \exists x(A(x)\land \neg B(x))}$
No A is B	AeB	${\displaystyle \neg \exists x(A(x)\land B(x))}$  or  ${\displaystyle \forall x(A(x)\rightarrow \neg B(x))}$
Some A is B	AiB	${\displaystyle \exists x(A(x)\land B(x))}$
Some A is not B	AoB	${\displaystyle \exists x(A(x)\land \neg B(x))}$

The convention here is that the letter S is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another classification of syllogisms known as the figure. Given that in each case the conclusion is S-P, the four figures are:

(Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan—reject the fourth figure as a figure distinct from the first.)

Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA below is AAA-1, or "A-A-A in the first figure".

The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are sometimes considered to commit the existential fallacy, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics. All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.

Fig. 1, treble clef. "A syllogism's letters can be best represented in music— take E, for example." -Marilyn Damord

The letters A, E, I, and O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc.

Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is the middle term, the second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P).

The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii) could also be written as "Some kittens are pets" (MiS in Datisi).

In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate ("logical not") the result of that expression.

It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms.[15]

Examples

M: men S: Greeks      P: mortal

Barbara (AAA-1)

   All men are mortal. (MaP)

   All Greeks are men. (SaM)

∴ All Greeks are mortal. (SaP)

M: reptile S: snake      P: fur

Celarent (EAE-1)

Similar: Cesare (EAE-2)

   No reptile has fur. (MeP)

   All snakes are reptiles. (SaM)

∴ No snake has fur. (SeP)

Camestres (AEE-2)
M: reptile S: fur      P: snake Camestres is essentially like Celarent with S and P exchanged. Similar: Calemes (AEE-4)    All snakes are reptiles. (PaM)    No fur bearing animal is a reptile. (SeM) ∴ No fur bearing animal is a snake. (SeP)

M: rabbit S: pet      P: fur

Darii (AII-1)

Similar: Datisi (AII-3)

   All rabbits have fur. (MaP)

   Some pets are rabbits. (SiM)

∴ Some pets have fur. (SiP)

Disamis (IAI-3)
M: rabbit S: fur      P: pet Disamis is essentially like Darii with S and P exchanged. Similar: Dimatis (IAI-4)    Some rabbits are pets. (MiP)    All rabbits have fur. (MaS) ∴ Some fur bearing animals are pets. (SiP)

M: homework S: reading      P: fun

Ferio (EIO-1)

Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)

   No homework is fun. (MeP)

   Some reading is homework. (SiM)

∴ Some reading is not fun. (SoP)

M: mammal S: pet      P: cat

Baroco (AOO-2)

   All cats are mammals. (PaM)

   Some pets are not mammals. (SoM)

∴ Some pets are not cats. (SoP)

M: cat S: mammal      P: pet

Bocardo (OAO-3)

   Some cats are not pets. (MoP)

   All cats are mammals. (MaS)

∴ Some mammals are not pets. (SoP)

---

M: man S: Greek      P: mortal

Barbari (AAI-1)

   All men are mortal. (MaP)

   All Greeks are men. (SaM)

∴ Some Greeks are mortal. (SiP)

Bamalip (AAI-4)
M: man S: mortal      P: Greek Bamalip is exactly like Barbari with S and P exchanged:    All Greeks are men. (PaM)    All men are mortals. (MaS) ∴ Some mortals are Greek. (SiP)

M: reptile S: snake      P: fur

Celaront (EAO-1)

Similar: Cesaro (EAO-2)

   No reptiles have fur. (MeP)

   All snakes are reptiles. (SaM)

∴ Some snakes have no fur. (SoP)

M: hooves S: human      P: horse

Camestros (AEO-2)

Similar: Calemos (AEO-4)

   All horses have hooves. (PaM)

   No humans have hooves. (SeM)

∴ Some humans are not horses. (SoP)

M: flower S: plant      P: animal

Felapton (EAO-3)

Similar: Fesapo (EAO-4)

   No flowers are animals. (MeP)

   All flowers are plants. (MaS)

∴ Some plants are not animals. (SoP)

M: square S: rhomb      P: rectangle

Table of all syllogisms

This table shows all 24 valid syllogisms, represented by Venn diagrams. Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an existential assumption are dashed.

figure	A ∧ A		A ∧ E				A ∧ I		A ∧ O		E ∧ I
1	Barbara	Barbari			Celarent	Celaront	Darii				Ferio
2			Camestres	Camestros	Cesare	Cesaro			Baroco		Festino
3		Darapti				Felapton	Datisi	Disamis		Bocardo	Ferison
4		Bamalip	Calemes	Calemos		Fesapo		Dimatis			Fresison

With Aristotle, we may distinguish singular terms, such as Socrates, and general terms, such as Greeks. Aristotle further distinguished types (a) and (b):

Such a predication is known as a distributive, as opposed to non-distributive as in Greeks are numerous. It is clear that Aristotle's syllogism works only for distributive predication, since we cannot reason All Greeks are animals, animals are numerous, therefore all Greeks are numerous. In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, Men can be predicated of Socrates but Socrates cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the terms must be general terms, or categorical terms as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that employ only categorical terms came to be called categorical syllogisms.

It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals. To argue that its validity can be explained by the theory of syllogism would require that we show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have existential import with respect to that term. It is ambiguous whether or not a universal statement of the form All A is B is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement All A is B has existential import with respect to A.

It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop "a companion-logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is 'a phrase signifying a thing's essence.'... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." [16]

However, many logic systems developed since do consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not supposit are false.

The following problems arise:

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

"All flying horses are mythical" is false if there are no flying horses.

If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC).

These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?

The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus, natural language statements—of the forms All A is B, No A is B, Some A is B, and Some A is not B—can be represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit or not made at all. Consequently, the four forms AaB, AeB, AiB, and AoB can be represented in first order predicate in every combination of existential import—so it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is no.

People often make mistakes when reasoning syllogistically.[17]

For instance, from the premises some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C.[18][19] However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black things (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions (C). This is because in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the "fallacy of the undistributed middle". Because of this, it can be hard to follow formal logic, and a closer eye is needed in order to ensure that an argument is, in fact, valid.[20]

Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.

In simple syllogistic patterns, the fallacies of invalid patterns are:

• Undistributed middle: Neither of the premises accounts for all members of the middle term, which consequently fails to link the major and minor term.
• Illicit treatment of the major term: The conclusion implicates all members of the major term (P – meaning the proposition is negative); however, the major premise does not account for them all (i.e., P is either an affirmative predicate or a particular subject there).
• Illicit treatment of the minor term: Same as above, but for the minor term (S – meaning the proposition is universal) and minor premise (where S is either a particular subject or an affirmative predicate).
• Exclusive premises: Both premises are negative, meaning no link is established between the major and minor terms.
• Affirmative conclusion from a negative premise: If either premise is negative, the conclusion must also be.
• Negative conclusion from affirmative premises: If both premises are affirmative, the conclusion must also be.

• Disjunctive syllogism
• Hypothetical syllogism
• Legal syllogism
• Polysyllogism
• Prosleptic syllogism
• Quasi-syllogism
• Statistical syllogism

• Syllogistic fallacy
• Argumentation theory
• Buddhist logic
• Enthymeme
• Formal fallacy
• Logical fallacy
• The False Subtlety of the Four Syllogistic Figures
• Tautology (logic)
• Venn diagram

• Smith, Robin. "Aristotle's Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Koutsoukou-Argyraki, Angeliki. Aristotle's Assertoric Syllogistic (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
• Lagerlund, Henrik. "Medieval Theories of the Syllogism". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Aristotle's Prior Analytics: the Theory of Categorical Syllogism an annotated bibliography on Aristotle's syllogistic
• Fuzzy Syllogistic System
• Development of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches
• Comparison between the Aristotelian Syllogism and the Indian/Tibetan Syllogism
• The Buddhist Philosophy of Universal Flux (Chapter XXIII – Members of a Syllogism (avayava))
• Online Syllogistic Machine An interactive syllogistic machine for exploring all the fallacies, figures, terms, and modes of syllogisms.